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A329050 Square array A(n,k) = prime(n+1)^(2^k), read by descending antidiagonals (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...; Fermi-Dirac primes (A050376) in matrix form, sorted into rows by their prime divisor.

%I A329050 #32 Feb 12 2020 02:11:06
%S A329050 2,4,3,16,9,5,256,81,25,7,65536,6561,625,49,11,4294967296,43046721,
%T A329050 390625,2401,121,13,18446744073709551616,1853020188851841,
%U A329050 152587890625,5764801,14641,169,17,340282366920938463463374607431768211456,3433683820292512484657849089281,23283064365386962890625,33232930569601,214358881,28561,289,19
%N A329050 Square array A(n,k) = prime(n+1)^(2^k), read by descending antidiagonals (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...; Fermi-Dirac primes (A050376) in matrix form, sorted into rows by their prime divisor.
%C A329050 This sequence is a permutation of A050376, so every positive integer is the product of a unique subset, S_factors, of its terms. If we restrict S_factors to be chosen from a subset, S_0, consisting of numbers from specified rows and/or columns of this array, there are notable sequences among those that may be generated. See the examples. Other notable sequences can be generated if we restrict the intersection of S_factors with specific rows/columns to have even cardinality. In any of the foregoing cases, the numbers in the resulting sequence form a group under the binary operation A059897(.,.).
%C A329050 Shares with array A246278 the property that columns grow downward by iterating A003961, and indeed, this array can be obtained from A246278 by selecting its columns 1, 2, 8, 128, ..., 2^((2^k)-1), for k >= 0.
%C A329050 A(n,k) is the image of the lattice point with coordinates X=n and Y=k under the inverse of the bijection f defined in the first comment of A306697. This geometric relationship can be used to construct an isomorphism from the polynomial ring GF(2)[x,y] to a ring over the positive integers, using methods similar to those for constructing A297845 and A306697. See A329329, the ring's multiplicative operator, for details.
%H A329050 Antti Karttunen, <a href="/A329050/b329050.txt">Table of n, a(n) for n = 0..77; the first 12 antidiagonals of array</a>
%H A329050 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polynomial_ring">Polynomial ring</a>
%F A329050 A(0,k) = 2^(2^k), and for n > 0, A(n,k) = A003961(A(n-1,k)).
%F A329050 A(n,k) = A182944(n+1,2^k).
%F A329050 A(n,k) = A329332(2^n,2^k).
%F A329050 A(k,n) = A225546(A(n,k)).
%F A329050 A(n,k+1) = A000290(A(n,k)) = A(n,k)^2.
%e A329050 The top left 5 X 5 corner of the array:
%e A329050   n\k |   0     1       2           3                   4
%e A329050   ----+-------------------------------------------------------
%e A329050    0  |   2,    4,     16,        256,              65536, ...
%e A329050    1  |   3,    9,     81,       6561,           43046721, ...
%e A329050    2  |   5,   25,    625,     390625,       152587890625, ...
%e A329050    3  |   7,   49,   2401,    5764801,     33232930569601, ...
%e A329050    4  |  11,  121,  14641,  214358881,  45949729863572161, ...
%e A329050 Column 0 continues as a list of primes, column 1 as a list of their squares, column 2 as a list of their 4th powers, and so on.
%e A329050 Every nonnegative power of 2 (A000079) is a product of a unique subset of numbers from row 0; every squarefree number (A005117) is a product of a unique subset of numbers from column 0. Likewise other rows and columns generate the sets of numbers from sequences:
%e A329050 Row 1:                 A000244 Powers of 3.
%e A329050 Column 1:              A062503 Squares of squarefree numbers.
%e A329050 Row 2:                 A000351 Powers of 5.
%e A329050 Column 2:              A113849 4th powers of squarefree numbers.
%e A329050 Union of rows 0 and 1:     A003586 3-smooth numbers.
%e A329050 Union of columns 0 and 1:  A046100 Biquadratefree numbers.
%e A329050 Union of row 0 / column 0: A122132 Oddly squarefree numbers.
%e A329050 Row 0 excluding column 0:  A000302 Powers of 4.
%e A329050 Column 0 excluding row 0:  A056911 Squarefree odd numbers.
%e A329050 All rows except 0:         A005408 Odd numbers.
%e A329050 All columns except 0:      A000290\{0} Positive squares.
%e A329050 All rows except 1:         A001651 Numbers not divisible by 3.
%e A329050 All columns except 1:      A252895 (have odd number of square divisors).
%e A329050 If, instead of restrictions on choosing individual factors of the product, we restrict the product to be of an even number of terms from each row of the array, we get A262675. The equivalent restriction applied to columns gives us A268390; applied only to column 0, we get A028260 (product of an even number of primes).
%t A329050 Table[Prime[#]^(2^k) &[m - k + 1], {m, 0, 7}, {k, m, 0, -1}] // Flatten (* _Michael De Vlieger_, Dec 28 2019 *)
%o A329050 (PARI)
%o A329050 up_to = 105;
%o A329050 A329050sq(n,k) = (prime(1+n)^(2^k));
%o A329050 A329050list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A329050sq(col, a-col))); (v); };
%o A329050 v329050 = A329050list(up_to);
%o A329050 A329050(n) = v329050[1+n];
%o A329050 for(n=0,up_to-1,print1(A329050(n),", ")); \\ _Antti Karttunen_, Nov 06 2019
%Y A329050 Transpose: A329049.
%Y A329050 Permutation of A050376.
%Y A329050 Rows 1-4: A001146, A011764, A176594, A165425 (after the two initial terms).
%Y A329050 Columns 1-5: A000040, A001248, A030514, A179645, A030635.
%Y A329050 Antidiagonal products: A191555.
%Y A329050 Subtable of A182944, A242378, A246278, A329332.
%Y A329050 A000290, A003961, A225546 are used to express relationship between terms of this sequence.
%Y A329050 Related binary operations: A059897, A306697, A329329.
%Y A329050 See also the table in the example section.
%K A329050 nonn,tabl
%O A329050 0,1
%A A329050 _Antti Karttunen_ and _Peter Munn_, Nov 02 2019
%E A329050 Example annotated for clarity by _Peter Munn_, Feb 12 2020